PHYS 102 Quiz 2 Core

[Ch 23] Electric potential V — what is it?

The electric potential energy per unit charge at a point. A scalar, measured in volts. 1 V = 1 J/C, and 1 V/m = 1 N/C.

[Ch 23] Potential difference — definition

ΔV = V_b − V_a = ΔU/q₀ = −∫(a to b) E·dl. Minus the work per unit charge the field does on a test charge. It depends only on the endpoints, never on the path.

[Ch 23] Link between potential and potential energy

U = qV, provided both are taken as zero at the same reference point.

[Ch 23] Which way does E point, and which way do charges move?

E points where V decreases fastest — downhill. A positive charge accelerates toward low V; a negative charge toward high V.

[Ch 23] Potential of a point charge

V = kq/r, with V = 0 at infinity. It carries the sign of q, so a negative charge gives a genuinely negative potential.

[Ch 23] Potential energy of two point charges

U = kq₀q/r. Positive if they repel, negative if they attract.

[Ch 23] Potential of several point charges

V = Σ kq_i/r_i — a plain scalar sum with signs kept. No components, no angles. This is why V is far easier to compute than E.

[Ch 23] Is V continuous? Is E?

V is continuous everywhere, except at a point or line charge. E is not — it jumps across a charged surface.

[Ch 23] Getting E back out of V

E_x = −dV/dx, E_r = −dV/dr, and in general E = −∇V. The field is minus the slope of the potential.

[Ch 23] What do flat and steep regions of a V-versus-x graph mean?

Flat means E = 0, however high the potential is. Steep means a strong field. If V rises with x, then E_x is negative.

[Ch 23] Potential of a uniformly charged spherical shell

V = kQ/r outside (r ≥ R), and V = kQ/R, a constant, inside (r ≤ R). Inside, E = 0 while V sits at its maximum.

[Ch 23] Field lines and equipotential surfaces

They always meet at 90 degrees, because a displacement along the surface gives dV = −E·dl = 0, so E has no component along it.

[Ch 23] What does the spacing of equipotentials tell you?

For a fixed potential difference between neighbouring surfaces, closer spacing means a stronger field.

[Ch 23] Potential of a conductor in equilibrium

Constant throughout the material and over its surface. The whole conductor is a single equipotential region.

[Ch 23] Electrostatic potential energy of a system of charges

U = ½ Σ q_i V_i, where V_i is the potential at charge i due to all the others. The ½ stops each pair being counted twice.

[Ch 23] Electrostatic potential energy of a charged conductor

U = ½QV. The first charge arrives free and the last costs the most, so the average potential during charging is half the final value.

[Ch 23] Does V = 0 imply E = 0, or E = 0 imply V = 0?

Neither. Midway between +q and −q, V = 0 but E is strongest. Inside a charged conductor E = 0 but V is at its maximum.

[Ch 23] Where does charge collect, and when does air break down?

Charge crowds where the radius of curvature is smallest, so the field is strongest at sharp points. Dry air breaks down at about 3 MV/m.

[Ch 24] Capacitance — definition and what fixes it

C = Q/V. It is set by the size, shape and arrangement of the conductors and by the medium between them — never by how much charge or voltage you put on it.

[Ch 24] Capacitance of a parallel-plate capacitor

C = ε₀A/d. More plate area raises C; a wider gap lowers it.

[Ch 24] Energy stored in a capacitor

U = ½QV = ½CV² = Q²/2C. Use whichever form matches the two quantities you already know.

[Ch 24] Why is the stored energy ½QV and not QV?

The voltage climbs as charge accumulates, so each dq costs (q/C)dq; integrating gives the triangle's area, ½Q²/C. The battery does the full QV — the other half is lost as heat.

[Ch 24] Energy density of an electric field

u = ½ε₀E², the energy per unit volume. True for any electric field, not just inside a capacitor.

[Ch 24] Capacitors in parallel

Same potential difference across each; the charges add. C_eq = C₁ + C₂ + … , always bigger than the largest one.

[Ch 24] Capacitors in series

Same charge on each; the potential differences add. 1/C_eq = 1/C₁ + 1/C₂ + … , always smaller than the smallest one.

[Ch 24] How do capacitor rules compare with resistor rules?

They are reversed. Capacitors add in parallel and go reciprocal in series; resistors add in series and go reciprocal in parallel. Sanity-check every answer against that.

[Ch 24] Dielectric constant κ

A dielectric weakens the field to E = E₀/κ and raises the capacitance to C = κC₀ = κε₀A/d. κ is always at least 1.

[Ch 24] Dielectric inserted: battery connected or disconnected?

Connected: V is pinned, C rises by κ, so Q rises by κ. Disconnected: Q is pinned, C rises by κ, so V and E both fall by κ. Either way C rises by κ.

[Ch 24] Why does a dielectric weaken the field inside it?

Molecular dipoles, permanent or induced, line up with the applied field and set up an opposing field. Uncancelled bound charge appears on the dielectric's surfaces.

[Ch 24] Values of ε₀ and k

ε₀ = 8.85 × 10⁻¹² C²/(N·m²), and k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C².

[Ch 25] Electric current

I = ΔQ/Δt, the rate of flow of charge through a cross-section. 1 A = 1 C/s.

[Ch 25] Current in terms of drift speed

I = qnAv_d, with n the carrier number density, A the cross-sectional area and v_d the drift speed — only a few mm/s in a real wire.

[Ch 25] Resistance and Ohm's law

R = V/I is the definition and always holds; 1 Ω = 1 V/A. V = IR is Ohm's law, and only applies where R stays constant — a straight line on a V-versus-I graph.

[Ch 25] Resistance in terms of resistivity

R = ρL/A. Longer means more resistance; thicker means less.

[Ch 25] Power in a resistor

P = IV = I²R = V²/R.

[Ch 25] What is emf?

The energy a source supplies per unit charge, measured in volts. It is not a force, and it is not the same thing as the terminal voltage.

[Ch 25] Terminal voltage and current for a real battery

V = ε − Ir, so the terminal voltage sags as the current rises. In a simple circuit, I = ε/(R + r).

[Ch 25] Resistors in series

The same current passes through each; the potential drops add. R_eq = R₁ + R₂ + … , bigger than the largest.

[Ch 25] Resistors in parallel

The same potential drop across each; the currents add. 1/R_eq = 1/R₁ + 1/R₂ + … , smaller than the smallest.

[Ch 25] Kirchhoff's two rules, and what each conserves

Junction rule: current in equals current out — conservation of charge. Loop rule: potential changes around any closed loop sum to zero — conservation of energy.

[Ch 25] The four loop-rule signs

Resistor along the current: −IR. Resistor against the current: +IR. Battery from − to +: +ε. Battery from + to −: −ε.

[Ch 25] Ammeter versus voltmeter

Ammeter: measures current, goes in series, needs near-zero resistance. Voltmeter: measures potential difference, goes in parallel, needs near-infinite resistance.

[Ch 25] Time constant of an RC circuit

τ = RC, in seconds. A larger R or a larger C makes everything slower.

[Ch 25] Discharging a capacitor

Q = Q₀e^(−t/τ), and the current decays the same way.

[Ch 25] Charging a capacitor

Q = Q_f(1 − e^(−t/τ)) with Q_f = Cε. The current still decays: I = I₀e^(−t/τ).

[Ch 25] How far along is an RC circuit after one time constant?

Charging is about 63 per cent complete; discharging leaves about 37 per cent. After roughly 5τ, treat it as finished.

[Ch 25] How does a capacitor behave at t = 0 and long afterwards?

At t = 0 an uncharged capacitor acts like a plain wire, so the current is at its maximum ε/R. Long afterwards it acts like a break in the circuit — no current, full voltage across it.

[Ch 26] Magnetic force on a moving charge

F = qv × B, perpendicular to both v and B.

[Ch 26] Magnitude of the magnetic force, and when it vanishes

F = |q|vB sin θ. Zero when v is parallel or antiparallel to B, and zero for a charge at rest.

[Ch 26] Right-hand rule for v × B

Fingers along v, curl them toward B, thumb gives v × B. That is the force if q is positive; reverse it if q is negative.

[Ch 26] Field units and page notation

1 T = 1 N/(A·m). A cross means the field points into the page (arrow tail feathers); a dot means out of the page (arrow tip).

[Ch 26] Do magnetic forces do work?

No. The force is always perpendicular to the velocity, so it changes direction only — never the speed and never the kinetic energy.

[Ch 26] Magnetic force on a current-carrying wire

F = IL × B, with L pointing along the current. For a bent wire, the net force equals that on a straight wire joining the two endpoints.

[Ch 26] Circular motion in a magnetic field — the starting point

Set the magnetic force equal to the centripetal force: qvB = mv²/r. Every other result in the section follows from this one line.

[Ch 26] Radius of a charged particle's circular orbit

r = mv/(qB), proportional to the momentum. More momentum means a wider circle.

[Ch 26] Cyclotron period and frequency

T = 2πm/(qB) and f = 1/T = qB/(2πm). Neither depends on speed or radius — only on q/m and B — so a faster particle traces a bigger circle in the same time.

[Ch 26] Motion when the velocity has a component along B

A helix. The parallel component drifts along the field line untouched; the perpendicular component goes in a circle.

[Ch 26] Velocity selector

Crossed E and B fields. Only particles travelling at v = E/B pass straight through, whatever their mass or charge.

[Ch 26] Mass spectrometer

m/q = B²r²/(2|ΔV|), from combining ½mv² = q|ΔV| in the accelerating gap with r = mv/(qB) in the field.

[Ch 26] Magnetic dipole moment of a current loop

μ = NIA n, where n is the unit normal found by curling your right hand in the direction of the current. Units A·m².

[Ch 26] Torque and net force on a current loop in a uniform field

τ = μ × B, of magnitude NIAB sin θ. The net force is zero, which is why a motor spins in place.

[Ch 26] Potential energy of a magnetic dipole

U = −μ·B = −μB cos θ, taken as zero at θ = 90 degrees.

[Ch 26] How do torque and energy vary with a dipole's orientation?

Aligned (0°): U = −μB lowest, τ = 0, stable. Perpendicular (90°): U = 0, τ = μB maximum. Opposed (180°): U = +μB highest, τ = 0, unstable.

[Ch 26] Hall effect

Carriers are shoved sideways by B until the electric field they build balances the magnetic force. V_H = IB/(nte). The sign of V_H reveals whether carriers are positive or negative; its size gives their number density.